Show that a non-abelain Lie algebra of dimension two has precisely three ideals.

Q: 7. Find all distinct principal ideals of Z14

A: Since Z14 are cyclic group with respect to addition so all ideals of Z14 are principal ideals

Q: Show that the union of a chain I1 ⊂ I2 ⊂ .... of ideals of a ring R isan ideal of R.

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Q: Determine whether T={(a, - a) | aEZ} is a subring of ZxZ.

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Q: Consider the ring ℤ12. List the elements of the following principal ideals: (i) ⟨4⟩  (ii) ⟨9⟩…

A: Given ring is ℤ12.

Q: Let S = {0, 2, 4, 6, 8} CZ10. Is S an ideal of Z10?

A:

Q: Q5:(A) Let R=(Z12 ,+12, -12 ). Find a) Maximal ideals of R b) Is (Z12, +12) a simple group? c) Is…

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Q: Let L be a finite dimensional Lie algebra. Any finite dimensional representation of L is completely…

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Q: Is it true or false that if I is a principal ideal in a commutative ring R with multiplicative…

A: Is it true or false that if  is a principal ideal in a commutative ring  with multiplicative…

Q: (B) Find all maximal ideals of (Z8, +8,8).

A:

Q: Show that the union of a chain I, CI,c of ideals of a ring R is an ideal of R. (This exercise is…

A: Given: We are given a chain of ideals in a ring Objective: The objective is to show that the union…

Q: Prove that gl(2, C) is isomorphic to the direct sum of sl(2, C) with C, the 1-dimensional complex…

A: gl(2,C):This is the Lie algebra of all 2×2 complex matrices. It is a 4-dimensional vector space over…

Q: Every submodule of a noetherian (artinian) module is noetherian (artinian).

A: Every submodule of a noetherian (artinian) module is noetherian (artinian).

Q: Is the ring Z/6Z and Z6 are isomorphic.

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Q: Compute the Killing form for the Heisenberg algebra and find its radical.

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Q: Prove that the extension of scalars from Z to the Gaussian integers Z[i] of the ring R is isomorphic…

A: It is given that ℤi is the Gaussian ring of integers. The objective is to prove that ℤi⊗ℤℝ≅ℂ. To…

Q: Show that for any k-algebras A1, A2,..., A,, we have (A1 x A2 x ...x A, )P S A" x A" x as algebras.

A: Let, A1,+,*,A2,+,*,...,Ar,+,* be the K-algebras, where + is the addition from 1 to r o which A1 is…

Q: Theorem: Let A and B be R-dibmodules af left R-modules M and Nrespectively. Then, MXN AXB

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Q: A proper ideal J of R is a maximal ideal if and only if R/J is a field.

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Q: Show that an intersection of ideals of a ring R is again an ideal of R.

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Q: rove that any unital irreducible R-module is cyclic.

A: Any unital irreducible R-module is cyclic

Q: Q2: Prove that the intersection of any two ideals of a ring R is also an ideal.

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Q: (a) Let S {C ): a, 6, c e Z, where 0 denotes the usual integer zero. Given that S is a ring under…

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Q: Show that a commutative ring R with unity is a field if and only if its only two ideals are (0) and…

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Q: The ring Z,3 has exactly-------------maximal ideals

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Q: Find the invariant factor decomposition of the Z-module HOM7(Z/30Z × Z/10Z, Z/15Z × Z/1OZ).

A: Given: HOMZ Z30Z×Z10Z,Z15Z×Z10Z

Q: Let L be a finite dimensional simple Lie algebra. Prove that any invariant bilinear form on L is a…

A: Let L be a finite-dimensional simple Lie algebra.  To prove any invariant bilinear form on L is a…

Q: Find the characteristic of the ring (Z/20Z) /(5Z/20Z)

A: We know that Z/nZ ring of integers modulo n has characteristic n. In our question ring (Z/20Z)…

Q: Theorem. Let M, N, and P be R-modules over a commutative ring 3.2 R. Then (i) MORN = NORM as…

A: note the theorem and I have used the notation of a function as an machine which operates  on an…

Q: Show that there is no associative algebra A such that the Lie algebra arising from A is sl2(C).

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Q: Is the factor ring Z5[x]/ a field?

A: